Pinball Dynamics: Unlimited Energy Growth in Switching Hamiltonian Systems

Maxim Arnold, Vadim Zharnitsky

Research output: Contribution to journalArticlepeer-review


A family of discontinuous symplectic maps arising naturally in the study of nonsmooth switched Hamiltonian systems is considered. This family depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving transformations due to nonlinear resonances. This paper provides a general description of the map and a construction of nontrivial unbounded solutions for the special case of the pinball transformation. An asymptotic expansion of the pinball map in the limit of large energy is derived and used for the construction of unbounded solutions. For the generic values of the parameters, in the large energy limit, the map behaves similarly to another one considered earlier by Kesten (Acta Arith 1966).

Original languageEnglish (US)
Pages (from-to)501-521
Number of pages21
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Sep 29 2015

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


Dive into the research topics of 'Pinball Dynamics: Unlimited Energy Growth in Switching Hamiltonian Systems'. Together they form a unique fingerprint.

Cite this